Software tools for stochastic programming: A Stochastic Programming Integrated Environment (SPInE)

SP models combine the paradigm of dynamic linear programming with modelling of random parameters, providing optimal decisions which hedge against future uncertainties. Advances in hardware as well as software techniques and solution methods have made SP a viable optimisation tool. We identify a growing need for modelling systems which support the creation and investigation of SP problems. Our SPInE system integrates a number of components which include a flexible modelling tool (based on stochastic extensions of the algebraic modelling languages AMPL and MPL), stochastic solvers, as well as special purpose scenario generators and database tools. We introduce an asset/liability management model and illustrate how SPInE can be used to create and process this model as a multistage SP application

Lagrangean decomposition for large-scale two-stage stochastic mixed 0-1 problems

In this paper we study solution methods for solving the dual problem corresponding to the Lagrangean Decomposition of two stage stochastic mixed 0-1 models. We represent the two stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangean Decomposition is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangean Decomposition schemes, as the Subgradient method, the Volume algorithm, the Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane scheme. We test the conditions and properties of convergence for medium and large-scale dimension stochastic problems. Computational results are reported.Progressive Hedging algorithm, volume algorithm, Lagrangean decomposition, subgradient method

Scenario trees and policy selection for multistage stochastic programming using machine learning

We propose a hybrid algorithmic strategy for complex stochastic optimization problems, which combines the use of scenario trees from multistage stochastic programming with machine learning techniques for learning a policy in the form of a statistical model, in the context of constrained vector-valued decisions. Such a policy allows one to run out-of-sample simulations over a large number of independent scenarios, and obtain a signal on the quality of the approximation scheme used to solve the multistage stochastic program. We propose to apply this fast simulation technique to choose the best tree from a set of scenario trees. A solution scheme is introduced, where several scenario trees with random branching structure are solved in parallel, and where the tree from which the best policy for the true problem could be learned is ultimately retained. Numerical tests show that excellent trade-offs can be achieved between run times and solution quality

Lagrangean decomposition for large-scale two-stage stochastic mixed 0-1 problems

In this paper we study solution methods for solving the dual problem corresponding to the Lagrangean Decomposition of two stage stochastic mixed 0-1 models. We represent the two stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangean Decomposition is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangean Decomposition schemes, as the Subgradient method, the Volume algorithm, the Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane scheme. We test the conditions and properties of convergence for medium and large-scale dimension stochastic problems. Computational results are reported.This research has been partially supported by the projects ECO2008-00777 ECON from the Ministry of Education and Science, Grupo de Investigación IT-347-10 from the Basque Government, grant FPU ECO-2006 from the Ministry of Education and Science, grants RM URJC-CM-2008-CET-3703 and RIESGOS CM from Comunidad de Madrid, and PLANIN MTM2009-14087-C04-01 from Ministry of Science and Innovation, Spain

Design and architecture of a stochastic programming modelling system

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Decision making under uncertainty is an important yet challenging task; a number of alternative paradigms which address this problem have been proposed. Stochastic Programming (SP) and Robust Optimization (RO) are two such modelling ap-proaches, which we consider; these are natural extensions of Mathematical Pro-gramming modelling. The process that goes from the conceptualization of an SP model to its solution and the use of the optimization results is complex in respect to its deterministic counterpart. Many factors contribute to this complexity: (i) the representation of the random behaviour of the model parameters, (ii) the interfac-ing of the decision model with the model of randomness, (iii) the difficulty in solving (very) large model instances, (iv) the requirements for result analysis and perfor-mance evaluation through simulation techniques. An overview of the software tools which support stochastic programming modelling is given, and a conceptual struc-ture and the architecture of such tools are presented. This conceptualization is pre-sented as various interacting modules, namely (i) scenario generators, (ii) model generators, (iii) solvers and (iv) performance evaluation. Reflecting this research, we have redesigned and extended an established modelling system to support modelling under uncertainty. The collective system which integrates these other-wise disparate set of model formulations within a common framework is innovative and makes the resulting system a powerful modelling tool. The introduction of sce-nario generation in the ex-ante decision model and the integration with simulation and evaluation for the purpose of ex-post analysis by the use of workflows is novel and makes a contribution to knowledge

Study Of Stochastic Market Clearing Problems In Power Systems With High Renewable Integration

Integrating large-scale renewable energy resources into the power grid poses several operational and economic problems due to their inherently stochastic nature. The lack of predictability of renewable outputs deteriorates the power grid’s reliability. The power system operators have recognized this need to account for uncertainty in making operational decisions and forming electricity pricing. In this regard, this dissertation studies three aspects that aid large-scale renewable integration into power systems. 1. We develop a nonparametric change point-based statistical model to generate scenarios that accurately capture the renewable generation stochastic processes; 2. We design new pricing mechanisms derived from alternative stochastic programming formulations of the electricity market clearing problem under uncertainty; 3. We devise a novel approach to coordinate strategic operations of multiple noncooperative system operators. The current industry practices are based on deterministic models that do not account for the stochasticity of renewable energy. Therefore, the solutions obtained from these deterministic models will not provide accurate measurements. Stochastic programming (SP) can accommodate the stochasticity of renewable energy by considering a set of possible scenarios. However, the reliability of the SP model solution depends on the accuracy of the scenarios. We develop a nonparametric statistical simulation method to develop scenarios for wind generation using wind speed data. In this method, we address the nonstationarity issues that come with wind-speed time-series data using a nonparametric change point detection method. Using this approach, we retain the covariance structure of the original wind-speed time series in all the simulated series. With an accurate set of scenarios, we develop alternative two-stage SP models for the two-settlement electricity market clearing problem using different representations of the non-anticipativity constraints. Different forms of non-anticipativity constraints reveal different hidden dual information inside the canonical two-stage SP model, which we use to develop new pricing mechanisms. The new pricing mechanisms preserve properties of previously proposed pricing mechanisms, such as revenue adequacy in expectation and cost recovery in expectation. More importantly, our pricing mechanisms can guarantee cost recovery for every scenario. Furthermore, we develop bounds for the price distortion under every scenario instead of the expected distortion bounds. We demonstrate the differences in prices obtained from the alternative mechanisms through numerical experiments. Finally, we discuss the importance of distributed smart grid operations inside the power grid. We develop an information and electricity exchange system among multiple distribution systems. These distribution systems participate/compete in common markets cohere electricity is exchanged. We develop a standard Nash game treating each distribution system (DS) as an individual player who optimizes their strategies separately. We develop proximal best response (BR) schemes to solve this problem. We present results from numerical experiments conducted on three and six DS settings

A Planner-Trader Decomposition for Multi-Market Hydro Scheduling

Peak/off-peak spreads on European electricity forward and spot markets are eroding due to the ongoing nuclear phaseout and the steady growth in photovoltaic capacity. The reduced profitability of peak/off-peak arbitrage forces hydropower producers to recover part of their original profitability on the reserve markets. We propose a bi-layer stochastic programming framework for the optimal operation of a fleet of interconnected hydropower plants that sells energy on both the spot and the reserve markets. The outer layer (the planner's problem) optimizes end-of-day reservoir filling levels over one year, whereas the inner layer (the trader's problem) selects optimal hourly market bids within each day. Using an information restriction whereby the planner prescribes the end-of-day reservoir targets one day in advance, we prove that the trader's problem simplifies from an infinite-dimensional stochastic program with 25 stages to a finite two-stage stochastic program with only two scenarios. Substituting this reformulation back into the outer layer and approximating the reservoir targets by affine decision rules allows us to simplify the planner's problem from an infinite-dimensional stochastic program with 365 stages to a two-stage stochastic program that can conveniently be solved via the sample average approximation. Numerical experiments based on a cascade in the Salzburg region of Austria demonstrate the effectiveness of the suggested framework

Numerical Methods for Scenario Tree Nonlinear Model Predictive Control

In this thesis we propose new methods in the field of numerical mathematics and stochastics for a model-based optimization method to control dynamical systems under uncertainty. In model-based control the model-plant mismatch is often large and unforeseen external influences on the dynamics must be taken into account. Therefore we extend the dynamical system by a stochastic component and approximate it by scenario trees. The combination of Nonlinear Model Predictive Control (NMPC) and the scenario tree approach to robustify with respect to the uncertainty is of growing interest. In engineering practice scenario tree NMPC yields a beneficial balance of the conservatism introduced by the robustification with respect to the uncertainty and the controller performance. However, there is a high numerical effort to solve the occuring optimization problems, which heavily depends on the design of the scenario tree used to approximate the uncertainty. A big challenge is then to control the system in real-time. The contribution of this work to the field of numerical optimization is a structure exploiting method for the large-scale optimization problems based on dual decomposition that yields smaller subproblems. They can be solved in a massively parallel fashion and additionally their discretization structure can be exploited numerically. Furthermore, this thesis presents novel methods to generate suitable scenario trees to approximate the uncertainty. Our scenario tree generation based on quadrature rules for sparse grids allows for scenario tree NMPC in high-dimensional uncertainty spaces with approximation properties of the quadrature rules. A further novel approach of this thesis to generate scenario trees is based on the interpretation of the underlying stochastic process as a Markov chain. Under the Markovian assumption we provide guarantees for the scenario tree approximation of the uncertainty. Finally, we present numerical results for scenario tree NMPC. We consider dynamical systems from the chemical industry and demonstrate that the methods developed in this thesis solve optimization problems with large scenario trees in real-time

A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming

Despite methodological advances for modeling decision problems under uncertainty, faithfully representing endogenous uncertainty still proves challenging, both in terms of modeling capabilities and computational requirements. A novel framework called Decision Programming provides an approach for solving such decision problems using off-the-shelf mathematical optimization solvers. This is made possible by using influence diagrams to represent a given decision problem, which is then formulated as a mixed-integer linear programming problem. In this paper, we focus on the type of endogenous uncertainty that received less attention in the introduction of Decision Programming: conditionally observed information. Multi-stage stochastic programming (MSSP) models use conditional non-anticipativity constraints (C-NACs) to represent such uncertainties, and we show how such constraints can be incorporated into Decision Programming models. This allows us to consider the two main types of endogenous uncertainty simultaneously, namely decision-dependent information structure and decision-dependent probability distribution. Additionally, we present a decomposition approach that provides significant computational savings and also enables considering continuous decision variables in certain parts of the problem, whereas the original formulation was restricted to discrete variables only. The extended framework is illustrated with two example problems. The first considers an illustrative multiperiod game and the second is a large-scale cost-benefit problem regarding climate change mitigation. Neither of these example problems could be solved with existing frameworks.Comment: 26 pages, 10 figure

Design and architecture of a stochastic programming modelling system

Decision making under uncertainty is an important yet challenging task; a number of alternative paradigms which address this problem have been proposed. Stochastic Programming (SP) and Robust Optimization (RO) are two such modelling ap-proaches, which we consider; these are natural extensions of Mathematical Pro-gramming modelling. The process that goes from the conceptualization of an SP model to its solution and the use of the optimization results is complex in respect to its deterministic counterpart. Many factors contribute to this complexity: (i) the representation of the random behaviour of the model parameters, (ii) the interfac-ing of the decision model with the model of randomness, (iii) the difficulty in solving (very) large model instances, (iv) the requirements for result analysis and perfor-mance evaluation through simulation techniques. An overview of the software tools which support stochastic programming modelling is given, and a conceptual struc-ture and the architecture of such tools are presented. This conceptualization is pre-sented as various interacting modules, namely (i) scenario generators, (ii) model generators, (iii) solvers and (iv) performance evaluation. Reflecting this research, we have redesigned and extended an established modelling system to support modelling under uncertainty. The collective system which integrates these other-wise disparate set of model formulations within a common framework is innovative and makes the resulting system a powerful modelling tool. The introduction of sce-nario generation in the ex-ante decision model and the integration with simulation and evaluation for the purpose of ex-post analysis by the use of workflows is novel and makes a contribution to knowledge.EThOS - Electronic Theses Online ServiceGBUnited Kingdo